Georgia Institute of Technology College of Sciences

Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields. They appear in nature: fireflies flashing in sync with each other is one fun situation.

In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneous Kuramoto systems (we will define these notions!). They are determined from a finite graph. In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations. We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems. We will see how computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions.

Note: we assume essentially NO dynamical systems nor algebraic geometry in this talk! This talk should be understandable to a general mathematical audience. The parts of the talk that are new represent joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend.

The spectrum of the discrete Laplacian on a infinite graph, or of its random perturbations such as the Anderson tight-binding model, encodes a rich information about the structure of that space. While natural questions abound (nature of the spectrum, localization of eigenfunctions, behavior of the spectral measure), few admit complete answers outside of very specific cases. In this talk, we will briefly survey some of the main open questions in the area. We will then present an elementary geometric criterion that provides control over the  regularity of the spectral measure.